t Let F be a real or complex n-dimensional map. It is said that F is globally
periodic if there exists some p ∈ N
+ such that Fp(x) = x for all x, where F
k = F ◦ F k−1, k ≥ 2. The minimal p satisfying this property is called the period of F. Given a m-dimensional parametric family of maps, say Fλ, a problem of current interest is to determine all the values of λ such that Fλ is globally periodic, together with their corresponding periods. The aim of this paper is to show some techniques that we
use to face this question, as well as some recent results that we have obtained. We
will focus on proving the equivalence of the problem with the complete integrability
of the dynamical system induced by the map F, and related issues; on the use of the
local linearization given by the Bochner Theorem; and on the use the Normal Form
theory. We also present some open questions in this setting.